The ProbForecastGOP Package

Transcription

1 The ProbForecastGOP Package April 24, 2006 Title Probabilistic Weather Field Forecast using the GOP method Version 1.3 Author Yulia Gel, Adrian E. Raftery, Tilmann Gneiting, Veronica J. Berrocal Description It produces probabilistic weather field forecasts using the Geostatistical Output Perturbation method introduced by Y. Gel, A. E. Raftery and T. Gneiting (JASA 99, 2004) Maintainer Veronica J. Berrocal Depends R (>= 1.8.0), fields, RandomFields License GPL version 2 or newer R topics documented: Emp.variog EmpDir.variog Field.sim ProbForecastGOP Variog.fit avg.variog avg.variog.dir forecast.grid gridlat gridlong linesmodel phase1temp plotfields slp Index 31 1

2 2 Emp.variog Emp.variog Empirical variogram of forecast errors averaged over time Description Calculates the empirical variogram of forecast errors, averaged over time. Usage Emp.variog(day, obs, forecast, id, coord1, coord2, cut.points=null, max.dist=nul Arguments day obs forecast id coord1 coord2 cut.points max.dist nbins numeric vector containing the day of observation. numeric vector containing the observed weather quantity. numeric vector containing the forecasted weather quantity. vector with the id of the metereological stations. vector containing the longitudes of the metereological stations. vector containing the latitudes of the metereological stations. numeric vector containing the cutpoints used for variogram binning. a numerical value giving the upper bound for the distance considered in the variogram computation. a numerical value giving the number of bins for variogram binning. If both cut.points and nbins are entered, the entry for nbins will be ignored and the vector with the cutpoints will instead be used for variogram binning. Details The function includes bias-correction; it regresses the forecasts on the observed weather quantity and computes the residuals. The empirical variogram of the residuals is then calculated by determining, for each day, the distance among all pairs of stations that have been observed in the same day and by calculating for each day the sum of all the squared differences in the residuals within each bin. These sums are then averaged over time, with weights for each bin given by the sum over time of the number of pairs of stations within the bin. The formula used is: γ(h) = d 1 2N (h,d) ( i (Y (x i + h, d) Y (x i, d)) 2 ) where γ(h) is the empirical variogram at distance h, N (h,d) is the number of pairs of stations that have been recorded at day d and whose distance is equal to h, and Y (x i + h, d) and Y (x i, d) are, respectively, the values of the residuals on day d at stations located at x i + h and x i. Variogram binning is ignored in this formula. - Defaults If the vector with the cutpoints is not specified, the cutpoints are determined so that there are nbins bins with approximately the same number of pairs per bin. If both the vector with the cutpoints and the number of bins, nbins, are unspecified, the function by default determines the cutpoints so that there are 300 bins with approximately the same number

3 Emp.variog 3 Value Note of pairs per bin. If both the vector with the cutpoints and the number of bins are provided, the entry for the number of bins is ignored and the vector with the cutpoints is used for variogram binning. The default value for the maximum distance considered in the variogram computation is the 90-th percentile of the distances between the stations. The function returns a list with components given by: mar.var Marginal variance of the forecast errors. bin.midpoints Numeric vector with midpoints of the bins used in the empirical variogram computation. number.pairs Numeric vector with the number of pairs per bin. empir.variog Numeric vector with the empirical variogram values. Depending on the data, the function might require substantial computing time. As a consequence, if the interest is in producing probabilistic weather forecasts and generating ensemble members, it is advised to save the output in a file and then use the Variog.fit and Field.sim functions. Author(s) Gel, Y., Raftery, A. E., Gneiting, T., Berrocal, V. J. veronica@stat.washington.edu. References Gel, Y., Raftery, A. E., Gneiting, T. (2004). Calibrated probabilistic mesoscale weather field forecasting: The Geostatistical Output Perturbation (GOP) method (with discussion). Journal of the American Statistical Association, Vol. 99 (467), See Also Cressie, N. A. C. (1993). Statistics for Spatial Data (revised ed.). Wiley: New York. EmpDir.variog for directional empirical variogram of forecast errors averaged over time, avg.variog and avg.variog.dir for, respectively, empirical and directional empirical variogram of a random variable averaged over time, and Variog.fit for estimation of parameters in a parametric variogram model. Examples ## Loading data data(slp) day <- slp$date.obs id <- slp$id.stat coord1 <- slp$lon.stat coord2 <- slp$lat.stat obs <- slp$obs forecast <- slp$forecast ## Computing variogram ## No specified cutpoints, no specified maximum distance ## Default number of bins

4 4 EmpDir.variog variogram <- Emp.variog(day=day,obs=obs,forecast=forecast,id=id, coord1=coord1,coord2=coord2,cut.points=null,max.dist=null,nbins=null) ## Plotting variogram plot(variogram$bin.midpoints,variogram$empir.variog,xlab="distance", ylab="semi-variance",main="empirical variogram") ## Computing variogram ## Specified cutpoints, specified maximum distance ## Unspecified number of bins variogram <- Emp.variog(day=day,obs=obs,forecast=forecast,id=id,coord1=coord1, coord2=coord2,cut.points=seq(0,1000,by=5),max.dist=800,nbins=null) ## Plotting variogram plot(variogram$bin.midpoints,variogram$empir.variog,xlab="distance", ylab="semi-variance",main="empirical variogram") EmpDir.variog Directional empirical variogram of forecast errors averaged over time Description Calculates directional empirical variogram of forecast errors, averaged over time. Usage EmpDir.variog(day, obs, forecast, id, coord1, coord2, tol.angle1=45, tol.angle2= Arguments day obs forecast id coord1 coord2 tol.angle1 tol.angle2 cut.points max.dist nbins type numeric vector containing the day of observation. numeric vector containing the observed weather quantity. numeric vector containing the forecasted weather quantity. vector with the id of the metereological stations. vector containing the longitudes of the metereological stations. vector containing the latitudes of the metereological stations. number giving a lower bound for the tolerance angle (measured in degrees). number giving an upper bound for the tolerance angle (measured in degrees). numeric vector containing the cutpoints used for variogram binning. a numerical value giving the upper bound for the distance considered in the variogram computation a numerical value giving the number of bins to use for variogram binning. If both cut.points and nbins are entered, the entry for nbins will be ignored and the vector with the cutpoints will instead be used for variogram binning. character string indicating the direction to use for variogram computations. Possible values are either E (for East-West) or N (for North-South).

5 EmpDir.variog 5 Details The function includes bias-correction; it regresses the forecasts on the observed weather quantity and computes the residuals. The directional empirical variogram of the residuals is then calculated by determining, for each day, the "directional" distance among all pairs of stations that have been observed in the same day and by calculating for each day the sum of all the squared differences in the residuals within each bin. These sums are then averaged over time, with weights for each bin given by the sum over time of the number of pairs of stations within the bin. The formula used is: γ(h) = d 1 2N (h,d) ( i (Y (x i + h, d) Y (x i, d)) 2 ) where γ(h) is the empirical variogram at distance h, N (h,d) is the number of pairs of stations that have been recorded at day d and whose distance is equal to h, and Y (x i + h, d) and Y (x i, d) are, respectively, the values of the residuals on day d at stations x i + h and x i. Variogram binning is ignored in this formula. The "directional" distance between two locations is defined to be equal to the distance between the two locations if the angle between the two locations is within the allowed range, while it is set equal to infinity if the angle between the two locations is outside the allowed range. - Defaults - By default, tol.angle1 and tol.angle2 are set to 45 and 135 degrees, respectively. If the vector with the cutpoints is not specified, the cutpoints are determined so that there are nbins number of bins with approximately the same number of pairs per bin. If both the vector with the cutpoints and the number of bins, nbins, are not provided, the function by default determines the cutpoints so that there are a total of 300 bins with approximately the same number of pairs per bin. If both the vector with the cutpoints and the number of bins are provided, the entry for the number of bins is ignored and the vector with the cutpoints is used for variogram binning. The default value for the maximum distance considered in the variogram computation is the 90-th percentile of the distances between the stations. Value The function returns a list with components given by: bin.midpoints Numeric vector with midpoints of the bins used in the directional empirical variogram computation. number.pairs Numeric vector with the number of pairs per bin. dir.variog Numeric vector with the directional empirical variogram values. Note The function might require some time to return an output. Author(s) Gel, Y., Raftery, A. E., Gneiting, T., Berrocal, V. J. veronica@stat.washington.edu.

6 6 Field.sim References Gel, Y., Raftery, A. E., Gneiting, T. (2004). Calibrated probabilistic mesoscale weather field forecasting: The Geostatistical Output Perturbation (GOP) method (with discussion). Journal of the American Statistical Association, Vol. 99 (467), Gel, Y., Raftery, A. E., Gneiting, T., Berrocal, V. J. (2004). Rejoinder. Journal of the American Statistical Association, Vol. 99 (467), See Also Cressie, N. A. C. (1993). Statistics for Spatial Data (revised ed.). Wiley: New York. Emp.variog for empirical variogram averaged over time, avg.variog and avg.variog.dir for, respectively, empirical and directional empirical variogram of a random variable averaged over time, and Variog.fit for estimation of parameters in a parametric variogram model. Examples ## Loading data data(slp) day <- slp$date.obs id <- slp$id.stat coord1 <- slp$lon.stat coord2 <- slp$lat.stat obs <- slp$obs forecast <- slp$forecast ## Computing directional variogram ## No specified cutpoints, no specified maximum distance ## No specified tolerance angles and default number of bins dir.variog <- EmpDir.variog(day,obs,forecast,id,coord1,coord2, tol.angle1=null,tol.angle2=null,cut.points=null,max.dist=null, nbins=null,type='e') ## Plotting directional variogram plot(dir.variog$bin.midpoints,dir.variog$dir.variog,xlab="distance", ylab="semi-variance",main="empirical Directional variogram") ## Computing directional variogram ## Specified cutpoints, specified maximum distance ## Specified tolerance angles and unspecified number of bins dir.variog <- EmpDir.variog(day,obs,forecast,id,coord1,coord2,tol.angle1=30, tol.angle2=150,cut.points=seq(0,1000,by=5),max.dist=800,nbins=null, type='n') ## Plotting directional variogram plot(dir.variog$bin.midpoints,dir.variog$dir.variog,xlab="distance",ylab="semi-variance", Field.sim Simulation of weather random field Description Simulates and displays realizations of forecast weather fields using a geostatistical model.

7 Field.sim 7 Usage Field.sim(obs, forecast, coord1.grid, coord2.grid, forecast.grid,variog.model="e qt.displ=c(10,50,90)) Arguments obs Details forecast coord1.grid numeric vector containing the observed weather quantity. numeric vector containing the forecasted weather quantity. numeric vector containing the longitudes of the grid points for the forecast. coord2.grid numeric vector containing the latitudes of the grid points for the forecast. forecast.grid numeric vector containing the forecast grid. variog.model character string with the name of the variogram model to be used for the simulations. Implemented models are exponential, spherical, gauss, matern, and gencauchy. param.est n.sim n.displ qt.displ numeric vector containing values for the parameters to use in the parametric variogram model. If the variog.model specified is exponential, spherical or gauss, then the parameters required are, in order, the nugget effect (a non-negative number), the variance and the range (both positive numbers). If the variog.model specified is gencauchy, the parameters required are, in order, the nugget effect (a non-negative number), the variance, the range (both positive numbers), the smoothness parameter a (a number in (0, 2]), and the long-range parameter b (a positive number). If the variog.model specified is matern the parameters required are, in order, the nugget effect (a non-negative number), the variance, the range, and the smoothness parameter a (all positive numbers). For more details on the equation of the variogram models listed above, look below at the section "Details". number of realizations to be simulated. number of realizations to be displayed on screen. numeric vector containing the quantiles to be displayed. The function initially estimates the additive and multiplicative bias by regressing the forecasts on the observed weather quantity. Then it simulates and displays realizations of the weather random field using the specified geostatistical model. Percentiles of the weather random field are calculated using the specified variogram model. - Parametric variogram models implemented - Below are the equations of the parametric variogram models that can be implemented. - exponential γ(d) = ρ + σ 2 (1 exp( d r )) where d is the distance, ρ is the nugget effect, σ 2 is the variance, and r is the range. - spherical

8 8 Field.sim γ(d) = ρ + σ 2 ( 3 2 d r 1 2 d3 r 3 ) where d is the distance, ρ is the nugget effect, σ 2 is the variance, and r is the range. - gauss Value γ(d) = ρ + σ 2 (1 exp( d2 r 2 )) where d is the distance, ρ is the nugget effect, σ 2 is the variance, and r is the range. - gencauchy γ(d) = ρ + σ 2 (1 (1 + da r a ) b a ) where d is the distance, ρ is the nugget effect, σ 2 is the variance, r is the range, a is the smoothness parameter, and b is the long-range parameter. - matern γ(d) = ρ + σ 2 (1 21 a Γ(a) da r a K a( d r )) where d is the distance, ρ is the nugget effect, σ 2 is the variance, r is the range, and a is the smoothness parameter. - Defaults- By default, 99 weather random fields are simulated, and 4 of them are displayed. If no vector of percentiles is provided, the 10th, 50th and 90th percentiles are determined and displayed. The function returns both a graphical and a numerical output. The numerical output is a list with the following components: model nugget variance Character string with the name of the variogram model used. Value of the nugget effect. Value of the variance. range Value of the rang. additional.par Value(s) of the additional parameters required and used in the variogram model. This is returned only if the variogram model used is matern or gencauchy sim.fields pct.fields 3-dimensional array where each layer contains the values of the simulated weather field at the gridded locations. 3-dimensional array where each layer contains the specified percentile. The graphical output consists in plots of the simulated weather fields displayed on multiple pages. Before displaying each page, the user will be asked for input.

9 Field.sim 9 Note This function depends on the following packages: RandomFields and fields. Author(s) Gel, Y., Raftery, A. E., Gneiting, T., Berrocal, V. J. veronica@stat.washington.edu. References Gel, Y., Raftery, A. E., Gneiting, T. (2004). Calibrated probabilistic mesoscale weather field forecasting: The Geostatistical Output Perturbation (GOP) method (with discussion). Journal of the American Statistical Association, Vol. 99 (467), Gneiting, T., Schlather, M. (2004). Stochastic models that separate the fractal dimension and the Hurst effect. SIAM Review 46, Schlather, M. (2001). Simulation and Analysis of Random Fields. R News 1(2), Nychka, D. (2004). The fields package. Available at: Stein, M. L. (1999). Interpolation of Spatial Data - Some Theory for Kriging. Springer-Verlag: New York. See Also plotfields, a plotting utility for weather random fields, GaussRF in the RandomFields package, for simulation of Gaussian random fields. Examples ## Loading data library(fields) library(randomfields) data(slp) data(gridlong) data(gridlat) data(forecast.grid) day <- slp$date.obs id <- slp$id.stat coord1 <- slp$lon.stat coord2 <- slp$lat.stat obs <- slp$obs forecast <- slp$forecast coord1.grid <- gridlong$gridded.lon coord2.grid <- gridlat$gridded.lat forecast.grid <- forecast.grid$gridded.forecast ## Computing the empirical variogram variogram <- Emp.variog(day=day,obs=obs,forecast=forecast,id=id, coord1=coord1,coord2=coord2,cut.points=null,max.dist=null,nbins=null) ## Estimating parameters param.est <- Variog.fit(emp.variog=variogram,variog.model="exponential", max.dist.fit=null,init.val=null,fix.nugget=false) ## Simulating realizations of the weather random field simul <- Field.sim(obs=obs,forecast=forecast,coord1.grid=coord1.grid, coord2.grid=coord2.grid,forecast.grid=forecast.grid,variog.model="exponential",

10 10 ProbForecastGOP param.est=c(param.est$nugget,param.est$variance,param.est$range),n.sim=4, n.displ=4,qt.displ=c(10,50,90)) ProbForecastGOP Probabilistic Weather Forecasts using the GOP method Description This function generates probabilistic forecasts of weather fields using the Geostatistical Output Perturbation method as described by Gel, Raftery and Gneiting (2004). Usage ProbForecastGOP(day, obs, forecast, id, coord1, coord2, cut.points=null, max.dist=null, nbins=300, variog.model="exponential", max.dist.fit=null, init.val=null, fix.nugget=false, coord1.grid, coord2.grid, forecast.grid, n.sim=99, out="sim", n.displ=4, qt.displ=c(10,50,90)) Arguments day obs forecast id coord1 coord2 cut.points max.dist nbins numeric vector containing the day of observation. numeric vector containing the observed weather quantity. numeric vector containing the forecasted weather quantity. vector with the id of the metereological stations. vector containing the longitudes of the metereological stations. vector containing the latitudes of the metereological stations. numeric vector containing the cutpoints used for variogram binning. a numerical value giving the upper bound for the distance considered in the variogram computation. a numerical value giving the number of bins for variogram binning. variog.model character string giving the name of the parametric model to fit to the empirical variogram. Implemented models are exponential, spherical, gauss, gencauchy, and matern. max.dist.fit number giving the maximum distance considered when fitting the variogram. init.val numeric vector giving the initial values for the parameters in the variogram model. The number of initial values to be entered depends on the parametric model specified. If the variog.model specified is exponential, spherical or gauss, then the parameters required are, in order, the nugget effect, the variance and the range. If the variog.model specified is gencauchy, the parameters required are, in order, the nugget effect, the variance, the range, the smoothness parameter a and the long-range parameter b. If the variog.model specified is matern the parameters required are, in order, the nugget effect, the variance, the range, and the smoothness parameter a. For more details on the valid range for the parameters and for the equation of the variogram models listed above, look below at the section "Details".

11 ProbForecastGOP 11 fix.nugget coord1.grid logical field indicating whether the nugget should be considered fixed or not. If TRUE the nugget effect will be assumed to be constant, and a value for the fixed nugget effect can be also provided. If the value provided is different from the one entered in the init.val field, then the value of the nugget effect is taken to be the one entered in the init.val field. If FALSE the nugget effect will be estimated along with the other parameters in the variogram model. numeric vector containing the longitudes of the grid points for the forecast field. coord2.grid numeric vector containing the latitudes of the grid points for the forecast field. forecast.grid numeric vector containing the forecast grid. n.sim out Details n.displ qt.displ number of realizations to be simulated. character string indicating the output produced by the function. Possible values are VARIOG, FIT and SIM. If VARIOG is specified, the function computes the empirical variogram of the forecast errors averaged over time and displays a plot of the empirical variogram. If FIT is specified, the function, in addition, fits a parametric model to the empirical variogram, returns estimates of the parameters and display both the empirical and the fitted variogram. If SIM is specified, the function simulates also realizations of the weather random field using the parametric variogram model specified. The function displays the simulated weather fields and some specified percentiles. number of realizations to be displayed on screen. numeric vector containing the quantiles to be displayed. This function generates probabilistic weather field forecasts by generating ensemble members using the Geostatistical Output Perturbation method presented by Gel, Raftery and Gneiting (2004). The Geostatistical Output Perturbation method consists of the following steps: 1. determination of the residuals, by regressing the forecasts on the observed weather quantity and computation of their empirical variogram. 2. estimation of the parameters of the variogram model fitted to the empirical variogram. 3. simulation of weather random fields using the estimated parametric variogram model. - Computation of the empirical variogram of the residuals The empirical variogram of the residuals is calculated by determining, for each day, the distance among all pairs of stations that have been observed in the same day and by calculating for each day the sum of all the squared differences in the residuals within each bin. These sums are then averaged over time, with weights for each bin given by the sum over time of the number of pairs of stations within the bin. The formula used is: γ(h) = d 1 2N (h,d) ( h (Y (i + h, d) Y (i, d)) 2 ) - Estimation of the parameters of the parametric variogram model fitted to the empirical variogram. The function estimates the parameters in the variogram model by minimizing the weighted leastsquare loss function

12 12 ProbForecastGOP LOSS(θ) = k ( ) 2 ˆγk γ k (θ) n k [ ] γ k (θ) where n k is the number of pairs contributing to the variogram computation in the kth bin, ˆγ k is the value of the empirical variogram in the kth bin and γ k (θ) is the value of the estimated parametric variogram model at the midpoint of the kth bin. - Parametric variogram models - The parametric variogram models implemented are: exponential, spherical, gaussian, generalized Cauchy and Whittle-Matern. - exponential The equation of the exponential variogram with parameters the nugget effect, ρ, the variance, σ 2, and the range, r, is given by: γ(d) = ρ + σ 2 (1 exp( d r )) where d is the distance between the two locations and γ(d) is the value of the exponential variogram at distance d. The nugget effect, ρ, is a non-negative number, while the variance, σ 2, and the range, r, are positive numbers. - spherical The equation of the spherical variogram with parameters the nugget effect, ρ, the variance, σ 2, and the range, r, is given by: γ(d) = ρ + σ 2 ( 3 2 d r 1 2 d3 r 3 ) where d is the distance between the two locations and γ(d) is the value of the spherical variogram at distance d. The nugget effect, ρ, is a non-negative number, while the variance, σ 2, and the range, r, are positive numbers. - gauss The equation of the gaussian variogram with parameters the nugget effect, ρ, the variance, σ 2, and the range, r, is given by: γ(d) = ρ + σ 2 (1 exp( d2 r 2 )) where d is the distance between the two locations and γ(d) is the value of the gaussian variogram at distance d. The nugget effect, ρ, is a non-negative number, while the variance, σ 2, and the range, r, are positive numbers. - gencauchy The equation of the generalized Cauchy variogram with parameters the nugget effect, ρ, the variance, σ 2, the range, r, the smoothness parameter, a, and the long-range parameter, b, is given by: γ(d) = ρ + σ 2 (1 (1 + da r a ) b a ) where d is the distance between the two locations and γ(d) is the value of the generalized Cauchy variogram at distance d. The nugget effect, ρ is a non-negative number, the variance, σ 2 and the

13 ProbForecastGOP 13 range, r, are positive numbers, the smoothness parameter, a, is a number in (0, 2], and the longrange parameter, b, is a positive number. - matern The equation of the Whittle-Matern variogram with parameters the nugget effect, ρ, the variance, σ 2, the range, r, and the smoothness parameter, a, is given by: Value γ(d) = ρ + σ 2 (1 ( 21 a Γ(a) da r a K a( d r )) where d is the distance between the two locations, γ(d) is the value of the Whittle-Matern variogram at distance d, Γ is the gamma function and K a is the Bessel function of the third kind with parameter a. The nugget effect, ρ, is a non-negative number, the variance, σ 2, the range, r, and the smoothness parameter, a, are positive numbers. - Defaults- If the vector with the cutpoints is not specified, the cutpoints are determined so that there are nbins bins with approximately the same number of pairs per bin. If both the vector with the cutpoints and the number of bins, nbins, are unspecified, the function by default determines the cutpoints so that there are 300 bins with approximately the same number of pairs per bin. The default value for the maximum distance considered in the variogram computation is the 90-th percentile of the distances between all stations. 1 The default value for the maximum distance is empirical variogram. 2 2 times the maximum distance considered in the Default for the initial values of the parameters is NULL, in which case the initial values for the parameters are determined internally and depend on the empirical variogram values. By default, the nugget effect is not considered constant. Thus, the default value for the fix.nugget field is FALSE. The default for out is SIM: by default, 99 weather random fields are simulated, and 4 of them are displayed. If no vector of percentiles, is provided, the 10th, 50th and 90th percentiles are determined and displayed. The function returns both a numerical and a graphical summary. The graphical and numerical summary provided depend on the specification given in the field out. The graphical summary consists of a plot of the empirical variogram, if out is equal to VARIOG. It consists of a plot of the empirical variogram with superimposed the fitted parametric model, if out is equal to FIT. If out is equal to SIM, the graphical summary consists of several plots: a plot of the empirical variogram with superimposed the fitted parametric model, plots of the simulated weather fields, and plots of specified percentiles of the weather random field. The plots are printed on multiple pages, and before displaying each page, the user will be asked for input. The numerical summary include the following: bias.coeff additive and multiplicative bias obtained by linear regression of the forecasted weather quantity on the observed weather quantity. Always provided. se.bias.coeff standard errors of the bias coefficients. Always provided. res.var variance of the forecast errors. Always provided.

14 14 ProbForecastGOP Note bin.midpoints numeric vector with midpoints of the bins used in the empirical variogram computation. Always provided. number.pairs numeric vector with the number of pairs per bin. Always provided. empir.variog numeric vector with the empirical variogram values. Always provided. model nugget variance name of the parametric model fitted to the empirical variogram. Provided only if out is equal to FIT or SIM. Estimate of the nugget effect. Provided only if out is equal to FIT or SIM. Estimate of the variance. Provided only if out is equal to FIT or SIM. range Estimate of the range. Provided only if out is equal to FIT or SIM. additional.par Numeric vector with the estimates of the additional parameters required by the parametric variogram model (only if the model specified is either whittlematern or gencauchy). This is provided only if out is equal to FIT or SIM. sim.fields pct.fields 3-dimensional array where each layer contains the values of the simulated weather field at the gridded locations. Provided only if out is equal to SIM. 3-dimensional array where each layer contains the specified percentiles of the weather field. Provided only if out is equal to SIM. Computation of the empirical variogram might require substantial computing time. Hence, it is advisable to run the function once with out equal to VARIOG, save part of the output and use it as input for the Variog.fit function to estimate a parametric variogram model. Finally, Field.sim will simulate realizations of the weather field from the specified geostatistical model. Author(s) Gel, Y., Raftery, A. E., Gneiting, T., Berrocal, V. J. veronica@stat.washington.edu. References Gel, Y., Raftery, A. E., Gneiting, T. (2004). Calibrated probabilistic mesoscale weather field forecasting: The Geostatistical Output Perturbation (GOP) method (with discussion). Journal of the American Statistical Association, Vol. 99 (467), Gel, Y., Raftery, A. E., Gneiting, T., Berrocal, V. J. (2004). Rejoinder. Journal of the American Statistical Association, Vol. 99 (467), Cressie, N. A. C. (1993). Statistics for Spatial Data (revised ed.). Wiley: New York. Gneiting, T., Schlather, M. (2004). Stochastic models that separate fractal dimension and the Hurst effect. SIAM Review, 46, Stein, M. L. (1999). Interpolation of Spatial Data - Some Theory for Kriging. Springer-Verlag: New York. See Also Schlather, M. (2001). Simulation and analysis of Random Fields. R News 1(2), Nychka, D. (2004). The fields package. Available at: Emp.variog for computation of empirical variogram of forecast errors averaged over time, Variog.fit for estimation of parameters in a geostatistical model using Weighted Least Square, Field.sim for realizations of weather fields with the specified covariance structure.

15 Variog.fit 15 Examples ## Loading data library(fields) library(randomfields) data(slp) day <-slp$date.obs id <- slp$id.stat coord1 <- slp$lon.stat coord2 <- slp$lat.stat obs <- slp$obs forecast <-slp$forecast data(gridlong) coord1.grid <- gridlong$gridded.lon data(gridlat) coord2.grid <- gridlat$gridded.lat data(forecast.grid) forecast.grid <- forecast.grid$gridded.forecast ## Specified cutpoints, default values for all the other fields. ## Only empirical variogram computation empirical.variog <- ProbForecastGOP(day=day,obs=obs,forecast=forecast, id=id,coord1=coord1,coord2=coord2,cut.points=seq(0,1000,by=2), max.dist=null,nbins=null,variog.model="exponential",max.dist.fit=null, init.val=null,fix.nugget=false,coord1.grid=coord1.grid, coord2.grid=coord2.grid,forecast.grid=forecast.grid,n.sim=10, out="variog",n.displ=4,qt.displ=c(5,50,95)) ## Unspecified cutpoints. ## Fit of a parametric variogram to an empirical variogram. fitted.variog <- ProbForecastGOP(day=day,obs=obs,forecast=forecast,id=id, coord1=coord1,coord2=coord2,cut.points=null,max.dist=null,nbins=null, variog.model="exponential",max.dist.fit=null,init.val=null,fix.nugget=false, coord1.grid=coord1.grid,coord2.grid=coord2.grid,forecast.grid=forecast.grid, n.sim=10,out="fit",n.displ=4,qt.displ=c(5,50,95)) ## Unspecified cutpoints. ## Whole routine. simulation.fields <- ProbForecastGOP(day=day,obs=obs,forecast=forecast,id=id,coord1=coord1, coord2=coord2,cut.points=null,max.dist=null,nbins=null,variog.model=null, max.dist.fit=null,init.val=null,fix.nugget=false,coord1.grid=coord1.grid, coord2.grid=coord2.grid,forecast.grid=forecast.grid,n.sim=4,out="sim", n.displ=4,qt.displ=c(5,50,95)) Variog.fit Fitting a parametric variogram model to an empirical variogram

16 16 Variog.fit Description This function estimates a parametric model fitted to an empirical variogram. The estimates are calculated using the weighted least squares method. Usage Variog.fit(emp.variog, variog.model="exponential", max.dist.fit=null, init.val=null, fix.nugget=false) Arguments Details emp.variog an object of the class emp.variog, output of the function Emp.variog or of the function avg.variog. variog.model character string giving the name of the parametric model to fit to the empirical variogram. Implemented models are exponential, spherical, gauss, gencauchy, and matern. max.dist.fit number giving the maximum distance considered when fitting the variogram. init.val fix.nugget numeric vector giving the initial values for the parameters. The number of initial values to be entered depends on the variogram model specified. If the variog.model specified is exponential, spherical or gauss, then the parameters required are, in order, the nugget effect, the variance and the range. If the variog.model specified is gencauchy, the parameters required are, in order, the nugget effect, the variance, the range, the smoothness parameter a, and the long-range parameter b. If the variog.model specified is matern the parameters required are, in order, the nugget effect, the variance, the range, and the smoothness parameter a. For more details on the valid range for the parameters and for the equation of the variogram models listed above, look below at the section "Details". logical field indicating whether the nugget should be considered fixed or not. If TRUE the nugget effect will be assumed to be constant, and a value for the fixed nugget effect can be also provided. If the value provided is different from the one entered in the init.val field, then the value of the nugget effect is taken to be the one entered in the init.val field. If FALSE the nugget effect will be estimated along with the other parameters in the variogram model. The function estimates the parameters in the variogram model by minimizing the weighted leastsquare loss function LOSS(θ) = k ( ) 2 ˆγk γ k (θ) n k [ ] γ k (θ) where n k is the number of pairs contributing to the variogram computation in the kth bin, ˆγ k is the value of the empirical variogram in the kth bin and γ k (θ) is the value of the estimated parametric variogram model at the midpoint of the kth bin. - Parametric variogram models -

17 Variog.fit 17 The parametric model implemented for the variogram are: exponential (exponential), spherical(spherical), gaussian (gauss), generalized Cauchy (gencauchy) and Whittle-Matern (matern). - exponential: The equation of the exponential variogram with parameters the nugget effect, ρ, the variance, σ 2, and the range, r, is given by: γ(d) = ρ + σ 2 (1 exp( d r )) where d is the distance between the two locations and γ(d) is the value of the exponential variogram at distance d. Notice that: the nugget effect, ρ, is a non-negative number, while the variance, σ 2, and the range, r, are positive numbers. - spherical The equation of the spherical variogram with parameters the nugget effect, ρ, the variance, σ 2, and the range, r, is given by: γ(d) = ρ + σ 2 ( 3 2 d r 1 2 d3 r 3 ) where d is the distance between the two locations and γ(d) is the value of the spherical variogram at distance d. Notice that: the nugget effect, ρ, is a non-negative number, while the variance, σ 2, and the range, r, are positive numbers. - gauss The equation of the gaussian variogram with parameters the nugget effect, ρ, the variance, σ 2, and the range, r, is given by: γ(d) = ρ + σ 2 (1 exp( d2 r 2 )) where d is the distance between the two locations and γ(d) is the value of the gaussian variogram at distance d. Notice that: the nugget effect, ρ, is a non-negative number, while the variance, σ 2, and the range, r, are positive numbers. - gencauchy The equation of the generalized Cauchy variogram with parameters the nugget effect, ρ, the variance, σ 2, the range, r, the smoothness parameter, a, and the long-range parameter, b, is given by: γ(d) = ρ + σ 2 (1 (1 + da r a ) b a ) where d is the distance between the two locations and γ(d) is the value of the generalized Cauchy variogram at distance d. Notice that: the nugget effect, ρ, is a non-negative number, the variance, σ 2, and the range, r, are positive numbers, the smoothness parameter, a, is a number in (0, 2], and the long-range parameter, b, is a positive number. - matern

18 18 Variog.fit Value The equation of the Whittle-Matern variogram with parameters the nugget effect, ρ, the variance, σ 2, the range, r, and the smoothness parameter, a, is given by: γ(d) = ρ + σ 2 (1 ( 21 a Γ(a) da r a K a( d r )) where d is the distance between the two locations, γ(d) is the value of the Whittle-Matern variogram at distance d, Γ is the gamma function and K a is the Bessel function of the third kind with parameter a. Notice that: the nugget effect, ρ, is a non-negative number, the variance, σ 2, the range, r, and the smoothness parameter, a, are positive numbers. - Defaults - 1 The default value for the maximum distance is empirical variogram. 2 2 times the maximum distance considered in the Default for the initial values of the parameters is NULL, in which case the initial values for the parameters are determined internally and depend on the empirical variogram values. By default, the nugget effect is not considered constant. Thus, the default value for the fix.nugget field is FALSE. The function returns a list with components given by: model nugget variance name of the parametric model fitted to the empirical variogram. Estimate of the nugget effect. Estimate of the variance. range Estimate of the range. additional.par Numeric vector with the estimates of the additional parameters required by the parametric variogram model. This is returned only if the parametric model fitted are gencauchy or matern Author(s) Gel, Y., Raftery, A. E., Gneiting, T., Berrocal, V. J. veronica@stat.washington.edu. References Gel, Y., Raftery, A. E., Gneiting, T. (2004). Calibrated probabilistic mesoscale weather field forecasting: the Geostatistical Output Perturbation (GOP) method (with discussion). Journal of the American Statistical Association, Vol. 99 (467), Cressie, N. A. C. (1993) Statistics for Spatial Data (revised ed.). Wiley: New York. Gneiting, T., Schlather, M. (2004). Stochastic models that separate fractal dimension and the Hurst effect. SIAM Review 46, Stein, M. L. (1999). Interpolation of Spatial Data - Some Theory for Kriging. Springer-Verlag: New York. See Also Emp.variog for computation of the empirical variogram of forecast errors, and avg.variog for computation of the empirical variogram of a random variable.

19 avg.variog 19 Examples ## Loading data library(fields) library(randomfields) data(slp) day <- slp$date.obs id <- slp$id.stat coord1 <- slp$lon.stat coord2 <- slp$lat.stat obs <- slp$obs forecast <- slp$forecast ## Computing the empirical variogram variogram <- Emp.variog(day=day,obs=obs,forecast=forecast,id=id,coord1=coord1,coord2=coord2, cut.points=null,max.dist=null,nbins=null) ## Estimating parameters ## Without specifying initial values for the parameters param.variog <- Variog.fit(emp.variog=variogram,variog.model="exponential",max.dist.fit=NULL, init.val=null,fix.nugget=false) ## Plotting the empirical variogram with the estimated parametric variogram superimposed plot(variogram$bin.midpoints,variogram$empir.variog,xlab="distance",ylab="semi-variance") lines(variogram$bin.midpoints,linesmodel(distance=variogram$bin.midpoints, variog.model="exponential",param=c(param.variog$nugget, param.variog$variance,param.variog$range))) ## Specifying the initial values for the parameters and keeping the nugget effect fixed param.variog <- Variog.fit(emp.variog=variogram,variog.model="exponential",max.dist.fit=NULL, init.val=c(0,2,100),fix.nugget=true) ## Plotting the empirical variogram with superimposed the estimated parametric variogram plot(variogram$bin.midpoints,variogram$empir.variog,xlab="distance",ylab="semi-variance") lines(variogram$bin.midpoints,linesmodel(distance=variogram$bin.midpoints, variog.model="exponential",param=c(param.variog$nugget, param.variog$variance,param.variog$range))) avg.variog Empirical variogram of a random variable averaged over time Description Calculates the empirical variogram of a random variable averaged over time. Usage avg.variog(day, coord1, coord2, id, variable, cut.points=null, max.dist=null, nb Arguments day numeric vector containing the day of observation.

20 20 avg.variog coord1 coord2 id variable cut.points max.dist nbins vector containing the longitudes of the metereological stations. vector containing the latitudes of the metereological stations. vector with the id of the metereological stations. numeric vector containing the variable for which the empirical varigram is to be computed. numeric vector containing the cutpoints used for variogram binning. a numerical value giving the upper bound for the distance considered in the variogram computation. a numerical value giving the number of bins for variogram binning. If both cut.points and nbins are entered, the entry for nbins will be ignored and the vector with the cutpoints will instead be used for variogram binning. Details The empirical variogram of the given random variable is calculated by determining, for each day, the distance among all pairs of stations that have been observed in the same day and by calculating for each day the sum of all the squared differences in the given random variable within each bin. These sums are then averaged over time, with weights for each bin given by the sum over time of the number of pairs of stations within the bin. The formula used is: γ(h) = d 1 2N (h,d) ( i (Y (x i + h, d) Y (x i, d)) 2 ) Value where γ(h) is the empirical variogram at distance h, N (h,d) is the number of pairs of stations that have been recorded at day d and whose distance is equal to h, and Y (x i + h, d) and Y (x i, d) are, respectively, the values of the given variable observed on day d at stations located at x i + h and x i. Variogram binning is ignored in this formula. - Defaults - If the vector with the cutpoints is not specified, the cutpoints are determined so that there are nbins bins with approximately the same number of pairs per bin. If both the vector with the cutpoints and the number of bins, nbins, are unspecified, the function by default determines the cutpoints so that there are 300 bins with approximately the same number of pairs per bin. If both the vector with the cutpoints and the number of bins are provided, the entry for the number of bins is ignored and the vector with the cutpoints is used for variogram binning. The default value for the maximum distance considered in the variogram computation is the 90-th percentile of the distances between the stations. The function returns a list with components given by: mar.var Marginal variance of the variable for which the empirical variogram is computed. bin.midpoints Numeric vector with midpoints of the bins used in the empirical variogram computation. number.pairs Numeric vector with the number of pairs per bin. empir.variog Numeric vector with the empirical variogram values.

21 avg.variog.dir 21 Note Depending on the data, the function might require substantial computing time. Author(s) Gel, Y., Raftery, A. E., Gneiting, T., Berrocal, V. J. References Gel, Y., Raftery, A. E., Gneiting, T. (2004). Calibrated probabilistic mesoscale weather field forecasting: The Geostatistical Output Perturbation (GOP) method (with discussion). Journal of the American Statistical Association, Vol. 99 (467), See Also Cressie, N. A. C. (1993). Statistics for Spatial Data (revised ed.). Wiley: New York. avg.variog.dir for directional empirical variogram of a random variable averaged over time, Emp.variog and EmpDir.variog for, respectively, empirical and directional empirical variogram of forecast errors averaged over time, and Variog.fit for estimation of parameters in a parametric variogram model. Examples ## Loading data data(slp) day <- slp$date.obs id <- slp$id.stat coord1 <- slp$lon.stat coord2 <- slp$lat.stat obs <- slp$obs forecast <- slp$forecast ## Computing variogram of observed temperature ## No specified cutpoints, no specified maximum distance ## Default number of bins variogram <- avg.variog(day=day, coord1=coord1,coord2=coord2,id=id,variable=obs,cut.point ## Plotting variogram plot(variogram$bin.midpoints,variogram$empir.variog,xlab="distance", ylab="semi-variance" avg.variog.dir Directional empirical variogram of a random variable averaged over time Description Usage Calculates directional empirical variogram of a random variable averaged over time. avg.variog.dir(day, coord1, coord2, id, variable, tol.angle1=45, tol.angle2=135, cut.points=null, max.dist=null, nbins=300, type)

22 22 avg.variog.dir Arguments day coord1 coord2 id variable tol.angle1 tol.angle2 cut.points max.dist nbins type numeric vector containing the day of observation. vector containing the longitudes of the metereological stations. vector containing the latitudes of the metereological stations. vector with the id of the metereological stations. numeric vector containing the variable for which the empirical variogram is to be computed. number giving a lower bound for the tolerance angle (measured in degrees). number giving an upper bound for the tolerance angle (measured in degrees). numeric vector containing the cutpoints used for variogram binning. a numerical value giving the upper bound for the distance considered in the variogram computation a numerical value giving the number of bins to use for variogram binning. If both cut.points and nbins are entered, the entry for nbins will be ignored and the vector with the cutpoints will instead be used for variogram binning. character string indicating the direction to use for variogram computations. Possible values are either E (for East-West) or N (for North-South). Details The directional empirical variogram of the given random variable is calculated by determining, for each day, the "directional" distance among all pairs of stations that have been observed in the same day and by calculating for each day the sum of all the squared differences in the given random variable within each bin. These sums are then averaged over time, with weights for each bin given by the sum over time of the number of pairs of stations within the bin. The formula used is: γ(h) = d 1 2N (h,d) ( i (Y (x i + h, d) Y (x i, d)) 2 ) where γ(h) is the empirical variogram at distance h, N (h,d) is the number of pairs of stations that have been recorded at day d and whose distance is equal to h, and Y (x i + h, d) and Y (x i, d) are, respectively, the values of the given variable observed on day d at stations located at x i + h and x i. Variogram binning is ignored in this formula. The "directional" distance between two locations is defined to be equal to the distance between the two locations if the angle between the two locations is within the allowed range, while it is set equal to infinity if the angle between the two locations is outside the allowed range. - Defaults - By default, tol.angle1 and tol.angle2 are set to 45 and 135 degrees, respectively. If the vector with the cutpoints is not specified, the cutpoints are determined so that there are nbins number of bins with approximately the same number of pairs per bin. If both the vector with the cutpoints and the number of bins, nbins, are not provided, the function by default determines the cutpoints so that there are a total of 300 bins with approximately the same number of pairs per bin. If both the vector with the cutpoints and the number of bins are provided, the entry for the number of bins is ignored and the vector with the cutpoints is used for variogram binning. The default value for the maximum distance considered in the variogram computation is the 90-th percentile of the distances between the stations.

23 avg.variog.dir 23 Value Note The function returns a list with components given by: bin.midpoints Numeric vector with midpoints of the bins used in the directional empirical variogram computation. number.pairs Numeric vector with the number of pairs per bin. dir.variog Numeric vector with the directional empirical variogram values. The function might require some time to return an output. Author(s) Gel, Y., Raftery, A. E., Gneiting, T., Berrocal, V. J. veronica@stat.washington.edu. References Gel, Y., Raftery, A. E., Gneiting, T. (2004). Calibrated probabilistic mesoscale weather field forecasting: The Geostatistical Output Perturbation (GOP) method (with discussion). Journal of the American Statistical Association, Vol. 99 (467), Gel, Y., Raftery, A. E., Gneiting, T., Berrocal, V. J. (2004). Rejoinder. Journal of the American Statistical Association, Vol. 99 (467), See Also Cressie, N. A. C. (1993). Statistics for Spatial Data (revised ed.). Wiley: New York. avg.variog for empirical variogram of a random variable averaged over time, Emp.variog and EmpDir.variog for, respectively, empirical and directional empirical variogram of forecast errors averaged over time, and Variog.fit for estimation of parameters in a parametric variogram model. Examples ## Loading data data(slp) day <- slp$date.obs id <- slp$id.stat coord1 <- slp$lon.stat coord2 <- slp$lat.stat obs <- slp$obs forecast <- slp$forecast ## Computing directional variogram of observed temperature ## No specified cutpoints, no specified maximum distance ## No specified tolerance angles and default number of bins dir.variog <- avg.variog.dir(day,coord1,coord2,id,variable=obs, tol.angle1=null,tol.angle ## Plotting directional variogram plot(dir.variog$bin.midpoints,dir.variog$dir.variog,xlab="distance", ylab="semi-variance"